Find the mean deviation about the mean for the following data : 6, 7, 10, 12, 13, 4, 8, 12.

Options

A. 3.76
B. 2
C. 2.75
D. 2.55

Answer

To find the mean deviation about the mean for the given data set \(6, 7, 10, 12, 13, 4, 8, 12\), we will follow these steps:

### Step 1: Calculate the Mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( \sum x_i \) is the sum of all observations and \( n \) is the number of observations.

**Calculating the sum of the observations:**
\[ 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72 \]

**Calculating the number of observations:**
There are 8 observations.

**Calculating the mean:**
\[ \bar{x} = \frac{72}{8} = 9 \]

### Step 2: Calculate \( |x_i - \bar{x}| \)
Next, we find the absolute deviations from the mean for each observation:
- For \( x_1 = 6 \): \( |6 - 9| = 3 \)
- For \( x_2 = 7 \): \( |7 - 9| = 2 \)
- For \( x_3 = 10 \): \( |10 - 9| = 1 \)
- For \( x_4 = 12 \): \( |12 - 9| = 3 \)
- For \( x_5 = 13 \): \( |13 - 9| = 4 \)
- For \( x_6 = 4 \): \( |4 - 9| = 5 \)
- For \( x_7 = 8 \): \( |8 - 9| = 1 \)
- For \( x_8 = 12 \): \( |12 - 9| = 3 \)

### Step 3: Sum of Absolute Deviations
Now we sum all the absolute deviations:
\[ 3 + 2 + 1 + 3 + 4 + 5 + 1 + 3 = 22 \]

### Step 4: Calculate Mean Deviation
Finally, we calculate the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]
Substituting the values we have:
\[ \text{Mean Deviation} = \frac{22}{8} = 2.75 \]

### Final Answer
The mean deviation about the mean for the given data is \( 2.75 \).

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